Final answer:
To solve Jack's work hours problem, establish the system of inequalities based on his hourly constraints, graph them, and find the feasible region. One possible solution that satisfies Jack's requirements of working at most 12 hours and no more than 8 hours tutoring is z = 5 and y = 7, assuming his income meets the minimum $180.
Step-by-step explanation:
To solve the system of inequalities related to Jack's work hours, we first need to establish the inequalities:
- z ≤ 8 (Jack works no more than 8 hours tutoring)
- y ≤ 12 - z (The total hours Jack can work is at most 12)
The next step is to determine the payment rates. Since the question does not specify, we will assume Jack earns the same rate for both jobs. Let's call this rate r.
To earn a minimum of $180, we can write the inequality as:
To graphically solve the system, we'll plot two lines representing the inequalities on a coordinate plane with z on the x-axis and y on the y-axis. The area that satisfies both inequalities will be the feasible region. We look for points within this region that also satisfy Jack's minimum earning requirement.
Regarding the possible solutions, we need to eliminate any answers that don't satisfy all conditions:
- z = 5 and y = 7: This solution satisfies both the condition of working at most 12 hours in total and tutoring for no more than 8 hours.
- z = 10 and y = 2: This solution does not satisfy the condition of tutoring for no more than 8 hours, so it cannot be correct.
Therefore, the correct and possible solution is z = 5 and y = 7, assuming the rate r is such that Jack earns at least $180 with these hours.